Strongly Convex Programming for Exact Matrix Completion and Robust Principal Component Analysis

The common task in matrix completion (MC) and robust principle component analysis (RPCA) is to recover a low-rank matrix from a given data matrix. Following the pioneering works of Cand`es et al., these problems have gained great attention recently from various areas in engineering and applied science. One fundamental result in MC and RPCA is that nuclear norm based convex optimizations guarantee exact low-rank matrix recovery. In this paper, we will show that strongly convex optimizations guarantee exact low-rank matrix recovery as well. The results in this paper not only provide sufficient conditions to the strongly convex models for exact low-rank matrix recovery, which in turn allows us to employ a broader range of existing optimization methods in the literature, but also guide us to choose suitable parameters in practical algorithms.